I imagine the kink to be the following image in the EM wave.
On the image, we see the charge movement 2 times and each one is very small movement. Even though it is small, it is easy to see why each movement produces the electric field a little bit shorter(due to finite speed of light).
What I dont understand is why the kink has the direction such as on the image.
I understand that electric field lines cant break but why does it join the vector as in the following image and what does join it ?
Electric field associated with moving charge shows what happens when a charge starts at rest and suddenly jerks to a new position and stops. This isn't very realistic, but it is easier to understand the field before and after. During is a little confusing. It would make more sense if the particle accelerated instead of instantly moved.
This link has an animation that gives a hint of why there are kinks. Electric field of a moving particle. Here is a screenshot from it.
This charge only behaves half as unreasonably. It starts out at rest, and suddenly begins moving at constant velocity of $80$% of the speed of light. If you drew the field lines, you would see a field lines pointed outward from the blue vectors, and then a field in different directions inside. There would be a kink at the bubble. We can use the animation to see what is going on.
The blue arrows show the field before it started. The yellow circle expands at the speed of light around the charge's original position.
At each new position, the field points away from the charge. The "news" of the each new position spreads outward at the speed of light. Points far away haven't heard about it yet. Even though the charge is moving, the news can't pass the original yellow bubble.
Points near the charge point away from it current position. Points farther away but inside the bubble point away from an older position. Inside the bubble, changes are smooth.
Changes are abrupt at the edge of the bubble because of the sudden start to the motion. It is a big change at the leading edge because the news very quickly changes from original position to very close. The field changes from medium strength to strong in a new direction. At the trailing edge, it is smaller because the change is from medium to weak in pretty much the same direction.
Here are some lecture notes that show in more detail how the kink works for the original case. Look at the diagrams near the bottom. It also shows field lines for a more reasonable acceleration.
